Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Average Error: 25.2 → 12.1

Time: 6.3s

Precision: 64

Math

\[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le 2.34703852665204316 \cdot 10^{195}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r795324 = x;
        double r795325 = y;
        double r795326 = r795325 - r795324;
        double r795327 = z;
        double r795328 = t;
        double r795329 = r795327 - r795328;
        double r795330 = r795326 * r795329;
        double r795331 = a;
        double r795332 = r795331 - r795328;
        double r795333 = r795330 / r795332;
        double r795334 = r795324 + r795333;
        return r795334;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r795335 = t;
        double r795336 = 2.347038526652043e+195;
        bool r795337 = r795335 <= r795336;
        double r795338 = x;
        double r795339 = y;
        double r795340 = r795339 - r795338;
        double r795341 = z;
        double r795342 = r795341 - r795335;
        double r795343 = a;
        double r795344 = r795343 - r795335;
        double r795345 = r795342 / r795344;
        double r795346 = r795340 * r795345;
        double r795347 = r795338 + r795346;
        double r795348 = r795338 * r795341;
        double r795349 = r795348 / r795335;
        double r795350 = r795339 + r795349;
        double r795351 = r795341 * r795339;
        double r795352 = r795351 / r795335;
        double r795353 = r795350 - r795352;
        double r795354 = r795337 ? r795347 : r795353;
        return r795354;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.2
Target	9.4
Herbie	12.1

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if t < 2.347038526652043e+195

   1. Initial program 22.0

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 22.0

      \[\leadsto x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{\color{blue}{1 \cdot \left(a - t\right)}}\]

   4. Applied times-frac 10.5

      \[\leadsto x + \color{blue}{\frac{y - x}{1} \cdot \frac{z - t}{a - t}}\]

   5. Simplified 10.5

      \[\leadsto x + \color{blue}{\left(y - x\right)} \cdot \frac{z - t}{a - t}\]

   if 2.347038526652043e+195 < t

   1. Initial program 51.6

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\]

   2. Taylor expanded around inf 25.4

      \[\leadsto \color{blue}{\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 2.34703852665204316 \cdot 10^{195}:\\ \;\;\;\;x + \left(y - x\right) \cdot \frac{z - t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x \cdot z}{t}\right) - \frac{z \cdot y}{t}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))